ASYMPTOTIC EXPANSION OF DIRAC-TYPE DISTRIBUTION ASSOCIATED WITH A CLASS OF HYPERSURFACES WITH DEGENERATED CRITICAL POINTS

In order to generalize Hadamard's theory of fundamental solutions to the case of degenerate holomorphic PDE, this paper studies the asymptotic expansion of Dirac-type distribution associated with a class of hypersurfaces F(x) with degenerate critical points and proves that [F(x)](+)(lambda) is a distribution-valued meromorphic of lambda is an element of C under some assumptions on F(x). Next, the authors use the Normal form theory of Arnold and prove that for a hypersurface F(x) = 0 with A(mu) type degenerate critical point at x = 0, F-+(lambda) is a distribution-valued meromorphic function of lambda.

Conformal Invariant Asymptotic Expansion Approach for Solving （3＋1）-Dimensional JM Equation

The （3＋1）-dimensional Jimbo-Miwa （JM） equation is solved approximately by using the conformal invariant asymptotic expansion approach presented by Ruan. By solving the new （3＋1）-dimensional integrable models, which are conformal invariant and possess Painlevé property, the approximate solutions are obtained for the JM equation, containing not only one-soliton solutions but also periodic solutions and multi-soliton solutions. Some approximate solutions happen to be exact and some approximate solutions can become exact by choosing relations between the parameters properly.

THE COMPLETE ASYMPTOTIC EXPANSION FOR BASKAKOV OPERATORS

In this paper, we derive the complete asymptotic expansion of classical Baskakov operators Vn （f;x） in the form of all coefficients of n^-k, k = 0, 1... being calculated explic- itly in terms of Stirling number of the first and second kind and another number G（i, p）. As a corollary, we also get the Voronovskaja-type result for the operators.

ASYMPTOTIC EXPANSIONS OF ZEROS FOR KRAWTCHOUK POLYNOMIALS WITH ERROR BOUNDS

Krawtchouk polynomials are frequently applied in modern physics. Based on the results which were educed by Li and Wong, the asymptotic expansions of Krawtchouk polynomials are improved by using Airy function, and uniform asymptotic expansions are got. Furthermore, the asymptotic expansions of the zeros for Krawtchouk polynomials are again deduced by using the property of the zeros of Airy function, and their corresponding error bounds axe discussed. The obtained results give the asymptotic property of Krawtchouk polynomials with their zeros, which are better than the results educed by Li and Wong.

A GLOBALLY UNIFORM ASYMPTOTIC EXPANSION OF THE HERMITE POLYNOMIALS

In this article,the author extends the validity of a uniform asymptotic expansion of the Hermite polynomials Hn（√2n＋1α）to include all positive values of α. His method makes use of the rational functions introduced by Olde Daalhuis and Temme （SIAM J.Math.Anal.,（1994）,25：304-321）.A new estimate for the remainder is given.

NOTE ON ASYMPTOTIC EXPANSION OF RIEMANN-SIEGEL INTEGRAL

In this note we establish two theorems concerning asymptotic expansion of Riemann-Siegel integrals as well as formula of generating function （double series） of coefficents of that expansion （for computation aims）; we also discuss similar results for Dirichlet series （L（s, fh） and L（s, X））, with m odd integer and X （ n ） （mod（ m ） ） （even） primitive characters （ inappendix B ） .

The Complete Asymptotic Expansion for the Baskakov-Kantorovich Operators

In this paper, we derive the Baskakov-Kantorovich operators Vn（f; x） the complete asymptotic expansion in the form of all coefficients of n^-k, k =0, 1 … being calculated explicitly. As a corollary, we also get the Voronovskaja-type result for the operators.

AN ASYMPTOTIC EXPANSION FORMULA OF KERNEL FUNCTION FOR QUASI FOURIER-LEGENDRE SERIES AND ITS APPLICATION

Is this paper we shall give cm asymptotic expansion formula of the kernel functim for the Quasi Faurier-Legendre series on an ellipse, whose error is 0(1/n2) and then applying it we shall sham an analogue of an exact result in trigonometric series.

THE ASYMPTOTIC EXPANSIONS OF SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS

In this paper we study the singularly penurbed boundary value problem: where e is a positive small parameter In the conditions: we prove the existences, and uniformly valid asymptotic expansions of solutions for the given boundary value problems, and hence we improve the existing results.

CERTAIN ASYMPTOTIC EXPANSIONS FOR LAGUERRE POLYNOMIALS AND CHARLIER POLYNOMIALS

Here proposed are certain asymptotic expansion formulas for Ln(w-1)(λz) and Cn(ω)(λz) in whichO(λ) and n = 0(λ1/2 )(λ→∞) , z being x complex number. Also presented are certain estimates for the remainders(error bounds) of the asymptotic expansions within the regions D1( - ∞<Rez≤1/2 (ω/λ) and D2(1/2 (ω/λ)≤Re.'C00)? respectively.

AN ASYMPTOTIC EXPANSION FORMULA FOR BERNSTEIN POLYNOMIALS ON A TRIANGLE

In this paper, an asymptotic expansion formula for approximation to a continuous function by Bernstein polymomials on a triangle is obtained.

Asymptotic Expansion of Standardized χ^(p)(n) Distribution and Power Function of χ^(p)(n)-Test

Suppose that Y follows a χ^(p)-distribution with n degrees of freedom, and Z is the standardized form of Y. Let f(z,n,p) and F(z,n,p) denote the density function and the distribution function of Z, respectively. In this paper, we obtain the asymptotic expansion for f(z,n,p) and F(z,n,p). The validity of these results is illuminated by some numerical examples. We also investigate the power function of χ^(p)-test by the asymptotic expansion.

The Time Asymptotic Expansion of the Bipolar Hydrodynamic Model for Semiconductors

In 2003, Gasser-Hsiao-Li [JDE(2003)] showed that the solution to the bipolar hydrodynamic model for semiconductors(HD model) without doping function time-asymptotically converges to the diffusion wave of the porous media equation(PME) for the switch-off case. Motivated by the work of Huang-Wu[arXiv:2210.13157], we will confirm that the time-asymptotic expansion proposed by Geng-Huang-Jin-Wu [arXiv:2202.13385] around the diffusion wave is a better asymptotic profile for the HD model in this paper, where we mainly adopt the approximate Green function method and the energy method.

Asymptotic expansions of complete Kahler-Einstein metrics with finite volume on quasi-projective manifolds

We give an elementary proof to the asymptotic expansion formula of Rochon and Zhang(2012)for the unique complete Kahler-Einstein metric of Cheng and Yau(1980),Kobayashi(1984),Tian and Yau(1987)and Bando(1990)on quasi-projective manifolds.The main tools are the solution formula for second-order ordinary differential equations(ODEs)with constant coefficients and spectral theory for the Laplacian operator on a closed manifold.

Asymptotic expansions of finite element solutions to Robin problems in H^3 and their application in extrapolation cascadic multigrid method

For the Poisson equation with Robin boundary conditions,by using a few techniques such as orthogonal expansion(M-type),separation of the main part and the finite element projection,we prove for the first time that the asymptotic error expansions of bilinear finite element have the accuracy of O(h3)for u∈H3.Based on the obtained asymptotic error expansions for linear finite elements,extrapolation cascadic multigrid method(EXCMG)can be used to solve Robin problems effectively.Furthermore,by virtue of Richardson not only the accuracy of the approximation is improved,but also a posteriori error estimation is obtained.Finally,some numerical experiments that confirm the theoretical analysis are presented.

ASYMPTOTIC RESULTS FOR OVER-DISPERSED OPERATIONAL RISK BY USING THE ASYMPTOTIC EXPANSION METHOD

In this paper, the author considers a new Loss-distribution-approach model, in which the over-dispersed operational risks are modeled by the compound negative binomial process. In the single dimensional case, asymptotic expansion for the quantile of compound negative binomial process is explored for computing the capital charge of a bank for operational risk. Moreover, when the dependence structure between different risk cells is modeled by the Frank copula, this approach is extended to the multi-dimensional setting. A practical example is given to demonstrate the effectiveness of approximation results.

Asymptotic Expansions of Transition Densities for Hybrid Jump-diffusions

A class of hybrid jump diffusions modulated by a Markov chain is considered in this work. The motivation stems from insurance risk models, and emerging applications in production planning and wireless communications. The models are hybrid in that they involve both continuous dynamics and discrete events. Under suitable conditions, asymptotic expansions of the transition densities for the underlying processes are developed. The formal expansions are validated and the error bounds obtained.

Asymptotic Expansions of Backward Equations for Two-time-scale Markov Chains in Continuous Time

This work develops asymptotic expansions for solutions of systems of backward equations of time- inhomogeneous Maxkov chains in continuous time. Owing to the rapid progress in technology and the increasing complexity in modeling, the underlying Maxkov chains often have large state spaces, which make the computa- tional tasks ihfeasible. To reduce the complexity, two-time-scale formulations are used. By introducing a small parameter ε〉 0 and using suitable decomposition and aggregation procedures, it is formulated as a singular perturbation problem. Both Markov chains having recurrent states only and Maxkov chains including also tran- sient states are treated. Under certain weak irreducibility and smoothness conditions of the generators, the desired asymptotic expansions axe constructed. Then error bounds are obtained.

Error Bounds for Uniform Asymptotic Expansions-Modified Bessel Function of Purely Imaginary Order

The authors modify a method of Olde Daalhuis and Temme for representing the remainder and coefficients in Airy-type expansions of integrals.By using a class of rational functions,they express these quantities in terms of Cauchy-type integrals;these expressions are natural generalizations of integral representations of the coe?cients and the remainders in the Taylor expansions of analytic functions.By using the new representation,a computable error bound for the remainder in the uniform asymptotic expansion of the modified Bessel function of purely imaginary order is derived.

SERIES PERTURBATIONS APPROXIMATE SOLUTIONS TO N-S EQUATIONS AND MODIFICATION TO ASYMPTOTIC EXPANSION MATCHED METHOD

A method that series perturbations approximate solutions to N-S equations with boundary conditions was discussed and adopted. Then the method was proved in which the asymptotic solutions of viscous fluid flow past a sphere were deducted. By the ameliorative asymptotic expansion matched method, the matched functions, are determined easily and the ameliorative curve of drag coefficient is coincident well with measured data in the case that Reynolds number is less than or equal to 40 000.